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Chapter 9

Chapter 9. Term Structure of Interest Rates. Business Cycle Patterns for the Term Structure. Yield. Declining. Rising. Maturity. Yield Curves. The most common yield curve shape is upward-sloping. Declining yield curves occur when interest rates are historically high.

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Chapter 9

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  1. Chapter 9 Term Structure of Interest Rates

  2. Business Cycle Patterns for the Term Structure Yield Declining Rising Maturity

  3. Yield Curves • The most common yield curve shape is upward-sloping. • Declining yield curves occur when interest rates are historically high. • Short-term interest rates are more variable than long-term interest rates.

  4. Historical evidence indicates two other empirical regularities of the term structure: • For maturities of six months and less, the yield curve has an upward slope most of the time. • The prices of long-term bonds are more variable than the prices of short-term bonds.

  5. Short maturities: [Small][Large] Long maturities: [Huge][Small] The duration effect dominates for long maturities.

  6. Segmented Markets Theory • Each maturity is a separate market. • There is no substitutability between maturities. • This could explain any yield curve shape.

  7. Increasing Liquidity Premium Rates Risk Premium  Maturity 1

  8. Assumptions • Investors prefer the shortest maturity. • Investors are risk averse. • Longer-term bonds are riskier. • Borrowers are ignored.

  9. What does it mean to be risk averse? • Risk neutral: utility per dollar is constant. • Risk averse: utility per dollar is decreasing. • Risk seeker : utility per dollar is increasing.

  10. Utility Functions Utility Increasing utility per $ $1 Constant utility per $ Risk Seeking $1 Risk Neutral Decreasing utility per $ $1 $1 $1 Risk Averse $1 $

  11. Actual utility functions may have risk averse and risk seeking sections. Utility $

  12. Money Substitute Theory • There are many large investors with temporarily excess funds. • In a short while, these investors will need these funds. • They will not invest in maturities longer than the date the funds are needed. • The price of short maturities are driven up by this large demand and their interest rates down.

  13. A very steep yield curve for short maturities is implied. y Maturity

  14. The Expectations Hypothesis • This theory says that forward interest rates are determined by interest rates anticipated to prevail at future dates. • Notation.

  15. Rates observed Points in Time 0 1 2 3 4 0 R0,1 f0,2 f0,3 f0,4

  16. Rates observed Points in Time 0 1 2 3 4 0 R0,1 f0,2 f0,3 f0,4 1 2 3 4 1 R1,1 f1,3 f1,4

  17. Elapsed Time and Spot and Forward Interest Rates Rates Observed Points in Time 0 1 2 3 4 0 R0,1 f0,2 f0,3 f0,4 1 2 3 4 1 R1,1 f1,3 f1,4 2 3 4 2 R2,1 f2,4

  18. Unbiased Expectations Hypothesis: Forward Rates Predict Future Spot Interest Rates Rates observed Points in Time 0 1 2 3 4 f0,2 f0,3 f0,4 R0,1 0 1 2 3 4 R1,1 1 2 3 4 R2,1 2 3 4 R3,1 3

  19. Probability R1, 1 Mean Forward Interest Rate Forward interest rate f0, 2 is the mean of the one-period spot rate one year from now. f0, 2 =E(R1, 1)

  20. Probability R2, 1 Mean Forward Interest Rate Forward interest rate f0, 3 is the mean of the one-period spot rate two years from now. f0, 3 =E(R2, 1)

  21. Example of a Flat Term Structure Points in Time 0 1 2 3 4 Expected futurespot rates 0.06 0.06 0.06 Forward rates f0,2 = 0.06 f0,3 = 0.06 f0,4 = 0.06 Spot rates R0,1 = 0.06 R0,2 = 0.06 R0,3 = 0.06 R0,4 = 0.06

  22. Example of a Rising Term Structure Points in Time 0 1 2 3 4 Expected futurespot rates 0.06 0.0804 0.10 Forward rates f0,2 = 0.06 f0,3 = 0.0804 f0,4 = 0.10 Spot rates R0,1 = 0.04 R0,2 = 0.05 R0,3 = 0.06 R0,4 = 0.0699

  23. Example of a Declining Term Structure Points in Time 0 1 2 3 4 Expected futurespot rates 0.07 0.06 0.05 Forward rates f0,2 = 0.07 f0,3 = 0.06 f0,4 = 0.05 Spot rates R0,1 = .1131 R0,2 = .0913 R0,3 = .0808 R0,4 = 0.073

  24. Example of a Humped Term Structure Points in Time 0 1 2 3 4 Expected futurespot rates 0.07 0.06 0.05 Forward rates f0,2 = 0.07 f0,3 = 0.06 f0,4 = 0.05 Spot rates R0,1 = 0.0501 R0,2 = 0.06 R0,3 = 0.06 R0,4 = 0.0575

  25. Combined Theory f0,2 = E[R1,1] + L2. f0,3 = E[R2,1] + L3, where L3 > L2. General case: f0,j = E[Rj-1,1] + Lj.

  26. Rising Term Structure and Constant Expected Interest Rates Points in Time 0 1 2 3 4 Expected future spot rates 0.05 0.05 0.05 Liquidity premium L2 = 0.01 L3 = 0.015 L4 = 0.02 Forward rates f0,2 = 0.06 f0,3 = 0.065 f0,4 =0.07 Spot rates R0,1 = 0.05 R0,2 = 0.055 R0,3 = 0.058 R0,4 = 0.061

  27. Rising Liquidity Premiums and a Declining Term Structure Points in Time 0 1 2 3 4 Expected future spot rates 0.04 0.03 0.02 Liquidity premium L2 = 0.01 L3 = 0.015 L4 = 0.02 Forward rates f0,2 = 0.05 f0,3 = 0.045 f0,4 = 0.04 Spot rates R0,1 = 0.055 R0,2 = 0.052 R0,3 = 0.05 R0,4 = 0.047

  28. P0,n = The price observed at Time 0 of strip (zero-coupon bond) maturing at Time n. P1,n-1 = The price observed at Time 1 of strip (zero-coupon bond) maturing at Time n. HPR = Holding period return.

  29. Arithmetic Approximation: If Unbiased Expectations Hypothesis holds: E[HPR] = R0,1.

  30. Two-period Case

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